Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method
The fractional calculus is useful in describing the natural phenomena with memory effect. This paper addresses the fractional form of Ambartsumian equation with a delay parameter. It may be a challenge to obtain accurate approximate solution of such kinds of fractional delay equations. In the litera...
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MDPI AG
2020-12-01
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| Online Access: | https://www.mdpi.com/2227-7390/8/12/2247 |
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| author | Weam Alharbi Sergei Petrovskii |
| author_facet | Weam Alharbi Sergei Petrovskii |
| author_sort | Weam Alharbi |
| collection | DOAJ |
| description | The fractional calculus is useful in describing the natural phenomena with memory effect. This paper addresses the fractional form of Ambartsumian equation with a delay parameter. It may be a challenge to obtain accurate approximate solution of such kinds of fractional delay equations. In the literature, several attempts have been conducted to analyze the fractional Ambartsumian equation. However, the previous approaches in the literature led to approximate power series solutions which converge in subdomains. Such difficulties are solved in this paper via the Homotopy Perturbation Method (HPM). The present approximations are expressed in terms of the Mittag-Leffler functions which converge in the whole domain of the studied model. The convergence issue is also addressed. Several comparisons with the previous published results are discussed. In particular, while the computed solution in the literature is physical in short domains, with our approach it is physical in the whole domain. The results reveal that the HPM is an effective tool to analyzing the fractional Ambartsumian equation. |
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| spelling | doaj.art-b8a4d4f3df634f898208618d94cc8cb42023-11-21T01:42:15ZengMDPI AGMathematics2227-73902020-12-01812224710.3390/math8122247Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation MethodWeam Alharbi0Sergei Petrovskii1Department of Mathematics, Faculty of Sciences, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi ArabiaDepartment of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UKThe fractional calculus is useful in describing the natural phenomena with memory effect. This paper addresses the fractional form of Ambartsumian equation with a delay parameter. It may be a challenge to obtain accurate approximate solution of such kinds of fractional delay equations. In the literature, several attempts have been conducted to analyze the fractional Ambartsumian equation. However, the previous approaches in the literature led to approximate power series solutions which converge in subdomains. Such difficulties are solved in this paper via the Homotopy Perturbation Method (HPM). The present approximations are expressed in terms of the Mittag-Leffler functions which converge in the whole domain of the studied model. The convergence issue is also addressed. Several comparisons with the previous published results are discussed. In particular, while the computed solution in the literature is physical in short domains, with our approach it is physical in the whole domain. The results reveal that the HPM is an effective tool to analyzing the fractional Ambartsumian equation.https://www.mdpi.com/2227-7390/8/12/2247Ambartsumian equationfractional derivativehomotopy perturbation methodMittag-Leffler function |
| spellingShingle | Weam Alharbi Sergei Petrovskii Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method Mathematics Ambartsumian equation fractional derivative homotopy perturbation method Mittag-Leffler function |
| title | Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method |
| title_full | Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method |
| title_fullStr | Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method |
| title_full_unstemmed | Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method |
| title_short | Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method |
| title_sort | numerical analysis for the fractional ambartsumian equation via the homotopy herturbation method |
| topic | Ambartsumian equation fractional derivative homotopy perturbation method Mittag-Leffler function |
| url | https://www.mdpi.com/2227-7390/8/12/2247 |
| work_keys_str_mv | AT weamalharbi numericalanalysisforthefractionalambartsumianequationviathehomotopyherturbationmethod AT sergeipetrovskii numericalanalysisforthefractionalambartsumianequationviathehomotopyherturbationmethod |